A Continuous Implementation of a Second-Variation Optimal Control Method for Space Trajectory Problems

The paper describes a continuous second-variation method to solve optimal control problems with terminal constraints where the control is defined on a closed set. The integration of matrix differential equations based on a second-order expansion of a Lagrangian provides linear updates of the control and a locally optimal feedback controller. The process involves a backward and a forward integration stage, which require storing trajectories. A method has been devised to store continuous solutions of ordinary differential equations and compute accurately the continuous expansion of the Lagrangian around a nominal trajectory. Thanks to the continuous approach, the method adapts implicitly the numerical time mesh and provides precise gradient iterates to find an optimal control. The method represents an evolution to the continuous case of discrete second-order techniques of optimal control. The novel method is demonstrated on bang–bang optimal control problems, showing its suitability to identify automatically optimal switching points in the control without insight into the switching structure or a choice of the time mesh. A complex space trajectory problem is tackled to demonstrate the numerical robustness of the method to problems with different time scales.     

Necessary and sufficient conditions for quadratic stabilizability of switched systems on non-uniform time domains

In this paper, we consider the quadratic stabilizability via state feedback for a particular class of switched systems that evolve on a non-uniform time domain by introducing time scales theory. The system considered switches between a continuous-time subsystem with variable lengths and a discrete-time subsystem with variable discrete step sizes. Necessary and sufficient conditions are derived to guarantee the quadratic stability of this class of switched systems via a switching state feedback law based on the existence of a common positive definite matrix satisfying the quadratic stabilizability condition by considering that the two subsystems are unstable. By state feedback, we mean that the switching among subsystems depends on the system states. Current results for this kind of state switching feedback control are derived only for switched systems evolving on a continuous time domain or a discrete time domain with fixed step’s size. These results are not applicable for the particular class of switched systems where there is a mixing between the continuous and discrete dynamics. This motivates the derivation of a new and more general state feedback control law for switched systems in this work. A numerical example illustrating the results is presented.     

Design of adaptive LQ regulators for MIMO systems based on multirate sampling of the plant output

In this paper, the certainty equivalence principle is used to combine a discrete-time adaptive law with a control structure derived from the linear-quadratic regulation problem. The proposed control structure is mainly based on multirate sampling of the output of the continuous-time plant under control and allows us to regulate the sampled closed-loop system, subject to a quadratic performance criterion, without making assumptions on the plant other than controllability and observability and the knowledge of two sets of structural indices. Using the proposed algorithm, the adaptive regulation problem is reduced to the determination of a fictitious static state feedback controller. Known techniques usually resort to the computation of fullorder adaptive state observers, thus introducing high-order exogenous dynamics in the control loop. Moreover, persistence of excitation and parameter convergence of the plant are provided without making any additional assumption on the system, as compared to known adaptive linear-quadratic regulation schemes.     

Sampled‐data reliable stabilization of T‐S fuzzy systems and its application

In this article, based on sampled‐data approach, a new robust state feedback reliable controller design for a class of Takagi–Sugeno fuzzy systems is presented. Different from the existing fault models for reliable controller, a novel generalized actuator fault model is proposed. In particular, the implemented fault model consists of both linear and nonlinear components. Consequently, by employing input‐delay approach, the sampled‐data system is equivalently transformed into a continuous‐time system with a variable time delay. The main objective is to design a suitable reliable sampled‐data state feedback controller guaranteeing the asymptotic stability of the resulting closed‐loop fuzzy system. For this purpose, using Lyapunov stability theory together with Wirtinger‐based double integral inequality, some new delay‐dependent stabilization conditions in terms of linear matrix inequalities are established to determine the underlying system's stability and to achieve the desired control performance. Finally, to show the advantages and effectiveness of the developed control method, numerical simulations are carried out on two practical models. © 2016 Wiley Periodicals, Inc. Complexity 21: 518–529, 2016     

Robust sampled‐data control of uncertain switched neutral systems with probabilistic input delay

This article focuses on the robust sampled‐data control for a class of uncertain switched neutral systems based on the average dwell‐time approach. In particular, the system is considered with probabilistic input delay using sampled state vectors, which are described by the stochastic variables with a Bernoulli distributed white sequence and time‐varying norm‐bounded uncertainties. By constructing a novel Lyapunov–Krasovskii functional which involves the lower and upper bounds of the delay, a new set of sufficient conditions are derived in terms of linear matrix inequalities for ensuring the robust exponential stability of the uncertain switched neutral system about its equilibrium point. Moreover, based on the stability criteria, a state feedback sampled‐data control law is designed for the considered system. Finally, a numerical example based on the water‐quality dynamic model for the Nile River is given to illustrate the effectiveness of the proposed design technique. © 2015 Wiley Periodicals, Inc. Complexity 21: 308–318, 2016     

Transition Manifolds of Complex Metastable Systems

We consider complex dynamical systems showing metastable behavior, but no local separation of fast and slow time scales. The article raises the question of whether such systems exhibit a low-dimensional manifold supporting its effective dynamics. For answering this question, we aim at finding nonlinear coordinates, called reaction coordinates, such that the projection of the dynamics onto these coordinates preserves the dominant time scales of the dynamics. We show that, based on a specific reducibility property, the existence of good low-dimensional reaction coordinates preserving the dominant time scales is guaranteed. Based on this theoretical framework, we develop and test a novel numerical approach for computing good reaction coordinates. The proposed algorithmic approach is fully local and thus not prone to the curse of dimension with respect to the state space of the dynamics. Hence, it is a promising method for data-based model reduction of complex dynamical systems such as molecular dynamics.     

Gradient Descent Approach to Optimal Mode Scheduling in Hybrid Dynamical Systems

This paper concerns the problem of optimally scheduling the sequence of dynamic response functions in nonlinear switched-mode hybrid dynamical systems. The control parameter has a discrete component and a continuous component, namely the sequence of modes and the duration of each mode, while the performance criterion consists of a cost functional on the state trajectory. The problem is naturally cast in the framework of optimal control. This framework has established techniques sufficient to address the continuous part of the parameter, but lacks adequate tools to consider the discrete element. To get around this difficulty, the paper proposes a bilevel hierarchical algorithm. At the lower level, the algorithm considers a fixed mode sequence and minimizes the cost functional with respect to the mode durations; at the upper level, it updates the mode sequence by using a gradient technique that is tailored to the special structure of the discrete variable (mode sequencing). The resulting algorithm is not defined on a single parameter space, but rather on a sequence of Euclidean spaces of increasing dimensions, an unusual setting for which there is no established notion of convergence. The paper suggests first a suitable definition of convergence based on the concepts of optimality functions; then, it proves that the proposed algorithm converges in that sense. Research supported in part by the National Science Foundation under Grant #0509064.     

Non‐Lipschitzian control algorithms: application to a nanofriction model

Recently we proposed a new feedback control algorithm for quantities describing global features of non‐linear dynamical systems. The performance of the algorithm, which is based on the concepts of non‐Lipschitzian dynamics and global targeting, has been successfully demonstrated for systems confined to one spatial dimension and for a specific targeted global quantity, namely the velocity of the centre of mass. In this paper we extend the scope of the non‐Lipschitzian control scheme to multi‐dimensional systems and different targeted quantities. We illustrate the efficiency of the non‐Lipschitzian feedback w.r.t. the ordinary (Lipschitzian) feedback, as well as the robustness and accuracy of the algorithm in a broad variety of control scenarios on the 2‐d Frenkel‐Kontorova model for nanofriction. Published in 2005 by John Wiley & Sons, Ltd.     

Adaptive Component-Wise Multiple-Try Metropolis Sampling

One of the most widely used samplers in practice is the component-wise Metropolis–Hastings (CMH) sampler that updates in turn the components of a vector-valued Markov chain using accept–reject moves generated from a proposal distribution. When the target distribution of a Markov chain is irregularly shaped, a “good” proposal distribution for one region of the state–space might be a “poor” one for another region. We consider a component-wise multiple-try Metropolis (CMTM) algorithm that chooses from a set of candidate moves sampled from different distributions. The computational efficiency is increased using an adaptation rule for the CMTM algorithm that dynamically builds a better set of proposal distributions as the Markov chain runs. The ergodicity of the adaptive chain is demonstrated theoretically. The performance is studied via simulations and real data examples. Supplementary material for this article is available online.     

Asymptotic linearization and stabilization for a class of nonlinear systems

This paper presents a nonlinear output feedback which asymptotically linearizes the class of nonlinear, continuous-time, affine in the control systems having relative degree equal to the state space dimension. Moreover, we show that any set of eigenvalues can be assigned for the asymptotic closed-loop linear system. The controller is based on a nonlinear observer, presented in a previous paper, and on the linearizing state feedback proposed by Isidori and computed in the estimated state. The main result obtained is equivalent to the separation theorem in the linear case.This work was supported by the Italian Ministry for University and Scientific and Technological Research.     

Tree-valued resampling dynamics Martingale problems and applications

The measure-valued Fleming–Viot process is a diffusion which models the evolution of allele frequencies in a multi-type population. In the neutral setting the Kingman coalescent is known to generate the genealogies of the “individuals” in the population at a fixed time. The goal of the present paper is to replace this static point of view on the genealogies by an analysis of the evolution of genealogies. We encode the genealogy of the population as an (isometry class of an) ultra-metric space which is equipped with a probability measure. The space of ultra-metric measure spaces together with the Gromov-weak topology serves as state space for tree-valued processes. We use well-posed martingale problems to construct the tree-valued resampling dynamics of the evolving genealogies for both the finite population Moran model and the infinite population Fleming–Viot diffusion. We show that sufficient information about any ultra-metric measure space is contained in the distribution of the vector of subtree lengths obtained by sequentially sampled “individuals”. We give explicit formulas for the evolution of the Laplace transform of the distribution of finite subtrees under the tree-valued Fleming–Viot dynamics.     

An entire strategy for precise system tracking control of a revolving thin flexural link,part I: Finite element modeling and direct tuning controller design

An entire control strategy including a design based model, controller design, and system output modification for a distributed parameter system is illuminated by application to feedback control of a revolving thin flexural link. In Part I, a very realizable actuator and a sensor, which uses a motor and a tachometer, are applied to design the control system. The finite element modeling and the state space representation are obtained for the purpose of control system analysis and computer simulation. Instead of relying on parameter identification subroutines, a controller design based on directly tuning the parameter of the gain makes the closed-loop absolutely stable and good for system tracking control. This control system design scheme is robust, insensitive to system parameter changes, and this algorithm cannot depend on traditionally priori knowledge such as the system dimension, exact model, or observer design. The performance included in the presence of all the high frequency dynamics can be effectively shown through the computer simulation, and one is led to speculate that this design scheme may perform quite well in the real world implementation.     

Modeling and Analysis of Modal Switching in Networked Transport Systems

We consider networked transport systems defined on directed graphs: the dynamics on the edges correspond to solutions of transport equations with space dimension one. In addition to the graph setting, a major consideration is the introduction and propagation of discontinuities in the solutions when the system may discontinuously switch modes, naturally or as a hybrid control. This kind of switching has been extensively studied for ordinary differential equations, but not much so far for systems governed by partial differential equations. In particular, we give well-posedness results for switching as a control, both in finite horizon open loop operation and as feedback based on sensor measurements in the system.     

Optimal Control of Iterative Solution Methods for Linear Systems of Equations

Iterative solution methods for linear systems of equations can be regarded as discrete-time control systems, for which a stabilizing feedback control has to be found. Well known algorithms such as GMRES(m) may exhibit unstable dynamics or sensitive dependence on initial conditions, thus preventing the algorithm to converge to the desired solution. Based on linear system feedback design techniques a new algorithm is proposed that does not suffer under such shortcomings. Global convergence to the desired solution is shown for any initial state. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Parameter estimation for partially observable systems subject to random failure

In this paper, we present a parameter estimation procedure for a condition‐based maintenance model under partial observations. Systems can be in a healthy or unhealthy operational state, or in a failure state. System deterioration is driven by a continuous time homogeneous Markov chain and the system state is unobservable, except the failure state. Vector information that is stochastically related to the system state is obtained through condition monitoring at equidistant sampling times. Two types of data histories are available — data histories that end with observable failure, and censored data histories that end when the system has been suspended from operation but has not failed. The state and observation processes are modeled in the hidden Markov framework and the model parameters are estimated using the expectation–maximization algorithm. We show that both the pseudolikelihood function and the parameter updates in each iteration of the expectation–maximization algorithm have explicit formulas. A numerical example is developed using real multivariate spectrometric oil data coming from the failing transmission units of 240‐ton heavy hauler trucks used in the Athabasca oil sands of Alberta, Canada. Copyright © 2012 John Wiley & Sons, Ltd.     

Design of hybrid regrouping PSO–GA based sub-optimal networked control system with random packet losses

In this paper, a new approach has been presented to design sub-optimal state feedback regulators over networked control systems with random packet losses. The optimal regulator gains, producing guaranteed stability are designed with the nominal discrete time model of a plant using Lyapunov technique which produces a few set of bilinear matrix inequalities (BMIs). In order to reduce the computational complexity of the BMIs, a genetic algorithm (GA) based approach coupled with the standard interior point methods for LMIs has been adopted. A regrouping particle swarm optimization based method is then employed to optimally choose the weighting matrices for the state feedback regulator design that gets passed through the GA based stability checking criteria i.e. the BMIs. This hybrid optimization methodology put forward in this paper not only reduces the computational difficulty of the feasibility checking condition for optimum stabilizing gain selection but also minimizes other time domain performance criteria like expected value of the set-point tracking error with optimum weight selection based LQR design for the nominal system.     

Resonance and chaos of micro and nano electro mechanical resonators with time delay feedback

The resonance and chaos of micro (nano) electro mechanical resonators with time delay feedback is concerned in the paper. Based on the experimental results, a lumped single degree-of-freedom (1DOF) model is studied and the effects of time delay displacement and velocity feedback on the system are investigated. In order to have a deep insight into the system, the amplitude frequency response curve of the system is firstly obtained using the multiple scales method. The Melnikov function method is then extended to the two time delay systems, and the analytically required condition for chaos was obtained. Finally, the fourth-order Runge–Kutta method, point-mapping method and spectrum diagram are used to simulate the evolution of the dynamic behavior of the time delay control system. Also, the stability of this time delay control system is studied thoroughly. The results show that time delay feedback is a good method for the control system and that reasonable selection of control system parameters can effectively suppress the vibration level for micro/nano-electro-mechanical resonator systems.     

Nonlinear set-valued boundary value problems with continuous feedback

We obtain sufficient conditions for the existence of at least one absolutely continuous solution of a nonlinear functional-differential inclusion in a finite-dimensional space with nonlinear set-valued functional boundary conditions. The set-valuedness of the dynamics may be due to the presence of a control. In addition, we consider the case in which the set-valuedness of the system dynamics and the boundary conditions is specified by inequalities. We assume the presence of a continuous feedback and impose the requirements of solvability of the open-loop problem. Statements of problems of this type arise in connection with the analysis of conflict-control systems.     

Chaos control for Willamowski–Rössler model of chemical reactions

Real systems evolving towards complex state encounter chaotic behavior. This behavior is very important in chemical processes or in biological structures because it defines the direction of the evolution of the system. From this point of view, the capability of deliberate control of these phenomena has a great practical impact despite the fact that it is very difficult; this is the reason why theoretical models are useful in these situations. In order to obtain chaos control in chemical reactions, the analysis of the dynamics of Willamowski–Rössler system involving the synchronization of two Minimal Willamowski–Rössler (MWR) systems based on the adaptive feedback method of control is presented in this work. As opposed to previous studies where in order to obtain synchronization 3 controllers were used, implying from a practical point of view the control of the concentrations of three chemical species, in this study we showed that the use of just one is sufficient which in practice is important as controlling the concentration of a single chemical species would be much easier. We also showed that the transient time until synchronization depends on initial conditions of two systems, the strength and number of the controllers and we attempted to identify the best conditions for a practical synchronization.